A conducting circular loop of areais placed perpendicular to a magnetic field which varies as If the resistance of the loop is, then the average thermal energy dissipated in the loop in one period is ______ J. [2026]

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Solution

Q.
A conducting circular loop of area $1.0 m^{2}$ is placed perpendicular to a magnetic field which varies as $B = \sin (100 t) Tesla .$ If the resistance of the loop is $100 Ω$ , then the average thermal energy dissipated in the loop in one period is ______ J. [2026]

1 $\frac{π}{2}$

2 $π$

3 $π^{2}$

4 $2 π$

Ans.
(2)

Area of the loop = $1 m^{2}$

$B = \sin (100 t)$

$∴$ $ϕ = B A = \sin (100 t)$

$∴$ $\frac{d ϕ}{d t} = 100 \cos (100 t)$

$∴$ $P = \frac{V^{2}}{R} = \frac{10^{4} \cos^{2} (100 t)}{100}$

$∴$ $Thermal energy dissipated in 1 time period$

$= \int_{0}^{T} P d t = \int_{0}^{T} 100 \cos^{2} (100 t) d t$

$T = \frac{2 π}{100} = \frac{π}{50} sec$

$∴$ $Q = 100 \int_{0}^{π / 50} \cos^{2} (100 t) d t$

$= 100 \int_{0}^{π / 50} \frac{1 + \cos (200 t)}{2} d t$

$= 100 [\frac{π}{100}] = π$

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